Throughout the paper, different ratios of the number of degrees of freedom in the SM and the mirror worlds appear. They need to be computed numerically and correspond to \(\mathcal {O}(1)\)  corrections that in large regions of the parameter space can be neglected. This
allows the derivation of simple analytical formulas for the different observables, such as the contributions to \(N_{\rm eff}\) , or the axion relic density. In this Appendix we review the definition of these quantities and compute them numerically as a function of both the SM and mirror\(_k\)  temperatures for different values of \(\gamma _k\) . To this aim, the tabulated effective number of degrees of freedom (for the SM case) as a function of temperature provided in Ref. [1]} will be used (see also
Fig. REF ).
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